The Savage Lakes trail is the best access to the Carter Creek drainage, however, it is possible to bushwhack from the road, where Carter Creek crosses under. There is a faint trail on the east side of the creek that in 2014, wandered into the forest and which quickly dissipated. From there, continue uphill through forest, rock outcrops & short cliffs, fallen logs and dead trees to intersect the Carter Lake trail and follow it to the lake. It took us about an hour with full packs to do this, plus it was exhausting. In retrospect, you're better off following the Savage Lakes trail to where the Carter Lake trail turns off to the left even though you must hike most of the way to the lower lake before reaching the intersection. Follow the Carter Lake trail down to Carter Lake on some broad switchbacks through forest. At the lake, the Trails Illustrated map suggests that the trail goes around the east side of the lake. This is false. Cross the lake outlet on the west end. You should pick up a trail there and there are some camp spots nearby. The trail heads up along the west side of the creek through forest and heavy vegetation. In under a half mile, at these coordinates, it intersects a trail coming in from Henderson Park: N 39° 22' 44.00 W 06° 32' 39.40". Continue NE past that intersection. We found it rather obscured by lush vegetation.
The Carter trail continues NE, staying on the west side of the creek and passing a small pond at 10,700 ft. A short distance past the pond, the trail crosses to the east side of the creek, below where the creek emerges from a small gorge. When it crosses over, it follows along rock outcrops above the creek until the creek exits the gorge at its upper end in a place of lower willows. Right here, it crosses back over to the west side: N 39° 22' 59.63 W 106° 32' 08.65". From there, continue to the first lake. A backpack campsite we used lies on the east side of the creek well before the lake if doing this summit as a backpack. There are other sites farther upstream. Continue past lakes, 2,3, 4, & 5, (see map) then work off trail over to lake #6. This is where the main climbing event begins.
On the west side of the lake, there's a steep, flower-strewn slope that angles up to a sloping bench above covered with larger blocks of rock. Work your way through the heavy vegetation to that bench, then angle right, avoiding boulders as best you can and working over to the bottom of a talus slope with low tundra that empties out below a notch in the ridge high above. Work your way up the steep slope to the notch. The last few feet to gain the notch has what some might regard as a couple 3rd class moves. Exposure is not bad. Earlier in the season, gaining this east-facing notch may require an ice axe and spikes of some kind.
Once at the notch, the summit is not far away. Turn left and head out across rock ledges working your way upward at every opportunity. The hiking is really Class 2+, but this is the most exposed part of the hike. The rock is generally solid and there are adequate ledges to make route-finding easy. Continue a diagonal track upward (SW) and eventually gain the rocky ridge to follow it to the summit. That summit overlooks the Lime Creek drainage and the Strawberry Lakes. It's an impressive view with Ribbed Peak in the distance and some high 12ers close by that some may want to consider bagging. If not continuing on to any of the closer summits. return as you came.
It may be possible to reach this same summit by packing up to Josephine Lake and then accessing the SW ridge above Josephine and following that ridge NE to UN 13,002. While we have been to Josephine Lake (an easy backpack), and up to the tundra ridge above, we have not actually followed that ridge to the summit of UN13,002. This may prove to be an easier route if the ridge does not exceed Class 2+. You may want to consult other sites to see if there's any information.
The entire Carter Drainage in its upper reaches is "classic Colorado." It's gorgeous, especially the highest lake. The vague at times trail continues to that highest lake and actually drops down toward Blodgett Lake where you could access several high 12ers. This central core to the Holy Cross wilderness is rich in water, lakes, wildlife and mesmerizing views.
Links to other information, routes & trip reports for this peak that may be helpful.